Hong Qian Department of Applied Mathematics University of Washington Seattle, WA , USA. September 22, Abstract

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1 Thermodynamics of the General Diffusion Process: Equilibrium Supercurrent and Nonequilibrium Driven Circulation with Dissipation arxiv: v2 [math-ph] 19 Sep 2015 Hong Qian Department of Applied Mathematics University of Washington Seattle, WA , USA September 22, 2015 Abstract Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation J ss (x and the gradient of stationary probability density f ss (x > 0. A sufficient and necessary condition for the orthogonality is a decomposition of the drift b(x = j(x + D(x ϕ(x where j(x = 0 and j(x ϕ(x = 0. Stationary processes with such Maxwell-Boltzmann equilibrium has an underlying conservative dynamics ẋ = j(x ( f ss (x 1 J ss (x, and a first integral ϕ(x ln f ss (x = const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and 1

2 an extended energy conservation law among an entire family of diffusion processes with different parameter α can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven ϕ- ascending steps and spontaneous ϕ-descending movements, alternated with iso-ϕ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics, with contradistinction to an earlier one [H. Qian et. al., J. Stat. Phys. 107, 1129 (2002]. 1 Introduction P. W. Anderson, J. J. Hopfield, and many other condensed matter physicists have all pointed out that emergent phenomena at each and every different scales actually obey different laws which require research that is just as fundamental in its nature as any other research [1, 2, 3]. The intricate behavior of a complex dynamic is particularly pronounced at a mesoscopic scale which contains too many individual bodies from a Newton-Laplacian perspective but not sufficient many, and often too strong an interaction and too heterogeneous, for the universal statistical laws such as central limit theorems and Gaussian processes to apply. This paper provides a didactic mathematical narrative of the general diffusion process, as a concrete model for complex stochastic nonlinear dynamics, in the light of two very different thermodynamic interpretations. The first one has its root in over-damped soft matters where sustained cyclic motions are considered a driven phenomenon accompanied with dissipation. The second one is motivated by under-damped and strongly correlated systems in which oscillatory motions are often the consequences of a conservative dynamics. Throughout the paper, classical thermodynamic terminologies are introduced with precise mathematical definitions in the framework of the general diffusion process. They are perfectly consistent with equilibrium statistical mechanics [4, 5] and mostly in accord with known notions in nonequilibrium statistical physics. Their ultimate validity, of course, are judged by the internal logic of the mathematics. Indeed, there is a growing awareness that, in order to fully develop a thermodynamic theory for mesoscopic nonequilibrium systems, its foundation has to be shifted away from the empirical notion of local equilibrium first formulated by the Brussel school, toward a mathematical theory. In the present work, it is the Markov dynamics [6, 7, 8]. 2

3 As expressed by some high-energy physicists the rest is chemistry [1]; one can indeed learn from chemistry a very powerful perspective on complex systems around us: First, classical chemistry distinguish itself from physics by quantifying a dynamical system in terms of species, individuals, and the numbers of individuals in a particular species, rather than tracking the detailed particle positions and velocities. This practice is consistent with many-body physics in which Eulerian rather than Lagrangian description of a fluid, and second quantization, are prefered. This was a fundamental insight of Boltzmann who, together with Maxwell, Gibbs, Smoluchowski, Einstein, and Langevin, paved the way to use stochastic mathematics as the proper language for quantifying complex systems. Second, while nonequilibrium thermodynamics in homogeneous systems usually deals with temperature and pressure gradients, nonequilibrium chemical thermodynamics often deals with isothermal, isobaric systems with all kinds of interesting phenomena, including animated living matters, under chemical potential differences. Chemical equilibrium is actually an isothermal, dynamic concept. Third, perhaps the most profound insight from chemistry, is the recognition of emerging discrete states, and transitions among them in molecules each one a nonlinear continuous manyatom system in its own right: Such a state is sufficiently stable against small perturbations of the underlying equations of atomic motion to be identified as a distinct chemical species. Such a transition is on an entirely different time scale; it necessarily crosses a barrier chemists called a transition state. The rare event can be quantified in terms of an exponentially distributed random time and the notion of a reaction coordinate or an order parameter. This is a great achievement in multi-scale modeling by separation of time scales. Finally, and possibly a deep idea from chemistry, is that stationary probability, as an emergent statistical entity, can actually be formulated as a law of force that quantifies collective motion of a system. Entropic force arises from mere probabilistic descriptions; and the concept of potential of mean force first articulated by J. G. Kirkwood in the theory of fluid mixtures [9] is the ultimate explanation of equilibrium free energy; and it is actually a conditional probability! Chemical potential difference can do mechanical work; it can power a Maxwell demon [10]. While the chemistry providing a perspective, the mathematical theory of stochastic processes 3

4 fully developed in the first half of 20th century provides a powerful analytical tool for representing complex dynamics. In fact, many key notions in chemistry echo important concepts in the theory of probability [11]. In recent years, a nonequilibrium steady state with positive entropy production that is consistent with over-damped soft matters and biochemical systems [12, 13] has been mathematically defined in terms of Markov processes [14, 15, 16]. Stochastic thermodynamics has emerged as a unifying theory of nonequilibrium statistical mechanics [17, 18, 19, 20]. Classical phase transition theory can be understood using elementary chemical master equations and stochastic differential equations with bistability in the limit of both time and system s size tending to infinities [21, 22]. Indeed, stochastic dynamics, which formalizes rapid stochastic and slower nonlinear dynamics and privides dual descriptions on both individual trajectories and ensemble probability distributions, seems to be a natural match for Anderson s hierarchical structure of sciencs generated by symmetry breakings [21, 23]. Even the Universe has become only one of the individuals of a multiverse [24]. This paper is structured as follows: In Sec. 2, results following the first perspective are summarized [6, 7, 16]. This is the main story line of the theory of stochastic thermodynamics which goes much further to fully explore trajectory-based thermodynamics and fluctuation theorems [17, 18, 19, 20]. Sec. 3 begins with introducing the notion of Maxwell-Boltzmann (MB equilibrium with non-dissipative supercurrent, and presents the defining characteristics of diffusion processes that possess an MB equilibrium with circulations: (i an orthogonality between stationary current J ss and the gradient of the stationary potential ϕ(x = ln f ss (x; (ii a decomposition of b(x = j(x D(x ϕ(x where j(x = 0 and j(x ϕ(x = 0. Sec. 4 investigates the emergent divergence-free vector field j(x, α from a family of diffusion processes with MB equilibrium: ẋ = j(x, α has a first integral ϕ(x, α. The Helmholtz theorem is applied to establish an extended energy conservation law h = h(σ B, α for the entire family of diffusion processes, among the ϕ-level sets of which h is the energy and σ B is the Boltzmann entropy. Sec. 5 studies diffusion processes that do not meet the orthogonality condition. We argue that the stationary process of such a system has both external driving force and dissipation, thus it is a nonequilibrium steady state within the framework of under-damped thermodynamics. Sec. 6 provides some discussions. 4

5 2 Diffusion processes with and without circulation We conresider a family of stochastic, diffusion process X αβ (t with a transition probability density function f αβ (x, t y that satisfies Fokker-Planck equation f αβ (x, t t = with non-local boundary condition and initial data ( β 1 D(x, α f αβ (x, t b(x, αf αβ (x, t, (1 R n f αβ (x, tdx = 1; f αβ (x, 0 = δ(x y, (2 in which x, y, b R n, and D is a n n positive definite matrix. The α is a continuous parameter that defines the family of related diffusion processes; and the β is a scaling parameter quantifying the magnitude of the noise. We shall assume that a positive, steady state probability density function exists lim f αβ(x, t y = fαβ(x, ss (3 t which is independent of y and statisfies the stationary Fokker-Planck equation β 1 D(x, α f ss αβ(x b(x, αf ss αβ(x J ss (x, J ss (x = 0, (4 under the same boundary condition in (2. For more discussions on the mathematical setup of this problem, see [6, 16]. The following facts are known under appropriate mathematical conditions. All discussions in Sec. 2 and Sec. 3 assume a fixed value of α, which we shall suppress until Sec Diffusion processes with detailed balance The system (1 is called detailed balanced if J ss (x = 0 x R n. This is true if and only if a ϕ(x exists such that D 1 (xb(x = ϕ(x. Then f ss (x = Z 1 (βe βϕ(x where ϕ(x is a potential energy of the system, and Z(β is a normalization factor: Z(β = e βϕ(x dx. (5 R n 5

6 Then the quantity F (t = 1 ( f(x, t f(x, t ln dx (6 β R f n ss (x = ϕ(x β ( 1 f(x, t ln f(x, tdx + β 1 ln Z(β, (7 R n in which we introduced the notion as the expected value with respect to time-dependent probability distribution f(x, t. The first term in (7 is the mean energy of the system at time t, and the term in the parenthesis is Gibbs-Shannon entropy. Therefore, it is natural to indentify the F (t as an instantaneous, generalized free energy of the dynamical system at time t. Actually, F (t is defined with respect to the equilibrium free energy of the system: F (t 0 and it is zero when f(x, t = f ss (x. Classical statistical mechanics of inanimate matters uses universal mechanical energy as a reference point; thus the equilibrium free energy is β 1 ln Z(β. As a function of time, it can be mathematically shown that df (t = J(x, t µ(x, t 0, (8 R n in which µ(x, t = ϕ(x + β 1 ln f(x, t, J(x, t = f(x, td(x µ(x, t. µ(x, t can and should be interpreted as a generalized chemical potential, or thermodynamic force, and J(x, t as the corresponding thermodynamic flux. F (t monotonically decreases until reaching its minimum df (t zero. In fact, e p (t β [16, 6]. One also has ds(t d 0 is called entropy production rate for the diffusion process ( f(x, t ln f(x, tdx = e p (t + β d ϕ(x, (9 R n the right-hand-side of which are entropy production, usually written as d is differential, and des which is not a total is the heat flux due to exchange with the environment [12]. Nonequilibrium entropy balance equation like (9 was first put forward phenomenologically by the Belgian thermodynamist de Donder, founder of the Brussels School [26, 27]. The shift from using state function In classical statistical mechanics, Newtonian mechanical energy is given a priori. Then the potential condition D 1 (xb(x = ϕ(x becomes the fluctuation-dissipation relation. It is an essential equation completing a phenomenological theory of equilibrium fluctuations in terms of a diffusion process. It has the same nature as the detailed balance condition in discrete-state Markov process models widely used in chemistry [25]. 6

7 df (t entropy to free energy, and the fact that e p (t β, reflect Helmhotz s contribution to the Second Law as free energy decreases for a canonical system, not entropy increases ; but the origin of decreasing free energy is still the same positive entropy production. The mathematical theory of stochastic processes offers additional insights for systems with detailed balance: A stationary stochastic trajectory X(t is time-reversible in a statistical sense [6, 16]. Therefore, anything accomplished through a sequence of events has an equal probability of being undone; nothing can be accomplished in an equilibrium dynamics. The linear operator on the right-hand-side of (1 is self-adjoint; there can be no oscillatory dynamics, only multiexponential decays. 2.2 Diffusion processes with unbalanced circulation A diffusion process without detailed balance has J ss (x 0, but J ss (x = 0. The system has unbalanced probability circulation in the stationary state as its hallmark. The two mathematical objects: ϕ β (x β 1 ln f ss β ss (x and Jβ (x can be understood in analogous to the potential and current in an electrical system. Note that ϕ β (x is now also a function of β, and its limit when β can be highly non-smooth. Still, if one introduces F (t as in (6, then Eq. (8 becomes df (t = E in (t e p (t, (10 where ( E in (t = J(x, t D 1 (xb(x β 1 ln fβ ss (x dx, (11 R n ( e p (t = J(x, t D 1 (xb(x β 1 ln f(x, t dx. (12 R n All three quantities have definitive sign [28, 29, 30, 31]: df (t 0, E in (t 0, e p (t 0. (13 Detailed balance holds if and only if E in (t = 0, stationarity holds if and only if df (t = 0, and e p (t = 0 implies both. There are two verbal interpretations for Eqs : (10 can be read as a generalized nonequilibrium free energy balance equation with instantaneous energy source df (t from its environment E in (t and dissipation e p (t [7, 8]. Alternatively, e p (t = + E in (t 7

8 can be read as total entropy production has two distinct origins: the spontaneous self-organization into stationary state and the continuous environmental drive that keeps the system away from its equilibrium. The two terms correspond nicely to Boltzmann s thesis and Prigogine s thesis on irreversibility, respectively. Quasi-steady state is a conceptual device that bridges these two views [32]. 2.3 Diffusion operator decomposition The right-hand-side of (1 is a second-order linear differential operator, L [ u ] = ( β 1 D(x, α u(x b(x, αu(x, (14 in an appropriate Hilbert space H, with inner product u, v = u(xv(x ( f ss (x 1 dx, u, v H. (15 R Then L is self-adjoint if and only if the diffusion process is detail balanced [6, 16]. Furthermore L = L S + L A, where u, v H. L S [u], v = u, L S [v], A diffusion process with self-adjoint L S has E in (t = 0. A degenerated diffusion with skew symmetric L A has L A [u], v = u, L A [v], df (t = 0 for all t. It actually has a nonrandom dynamics whose trajectories follow the ordinary differential equation ẋ = ( f ss β (x 1 J ss β (x [7]. The solution curves of this equation in phase space is identical to ẋ = J ss β Paradoxically, such a dynamical system is called conservative in classical mechanics. ss (x, Jβ (x = 0. Because the diffusion with L A is degenerate, both E in (t and e p (t in (11 and (12 are infinite thus no longer defined. The mathematics of decomposing L is not new per se [33, 34], but its clear relation with nonequilibrium thermodynamics and the theory of entropy production is novel [7, 8]. 8

9 3 ϕ J ss : Circulation as a supercurrent in a Maxwell- Boltzmann equilibrium All the mathematical narritive so far fits established chemical thermodynamics of over-damped molecular systems. In particular, when applied to a molecular motor, the E in term in (10 is indeed the amount of ATP hydrolysis free energy, and e p the heat dissipation [35]. In physics, however, the notion of a persistent current describes a perpetual electrical current without requiring an external power source. A superconducting current is one example. We now show that an alternative, novel thermodynamic interpretation based on an under-damped dynamic perspective is equally legitimate [8]: J ss (x = 0 is no longer the defining characteristics of an equilibrium. Rather we define a Maxwell-Boltzmann (MB equilibrium as J ss (x ϕ(x = 0 where ϕ(x = β 1 ln fβ ss (x. Unbalanced circulation on an equal-ϕ level set is considered conservative. The following facts are known under appropriate mathematical conditions. 3.1 ϕ(x is independent of β We assume, when β = 1, J ss 1 (x ϕ 1 (x. This implies also an orthogonality between J ss 1 (x and f ss 1 (x x R n. One can decompose vector field b(x as which can be re-written as b(x = ( f ss 1 (x 1 J ss 1 (x + D(x ln f ss 1 (x, (16 b(x = ( f1 ss (x ( β ( f ss 1 (x 1+β J ss 1 (x + β 1 D(x ln ( f1 ss (x β. (17 Since ( (f ss 1 (x 1+β J ss 1 (x = 0, we identify ( f1 ss (x 1+β J ss 1 (x = Jβ ss(x and ( f1 ss (x β = f ss(x, which is a solution to (4. Therefore, f ss β (x = Z 1 (βe βϕ(x. Furthermore, J ss β (x = j(xe βϕ(x in which j(x is also independent of β and divergence free: j(x = e βϕ(x J ss β (x + J ss β (x e βϕ(x = 0. 9 β

10 The diffusion process in (1 has an MB equilibrium if and only if [8] b(x = j(x D(x ϕ(x, j(x = 0, j(x ϕ(x = 0. (18 This result generalizes the potential condition in Sec. 2.1 with an additional divergence-free, orthogonal j(x. (18 is a much more restrictive condition on b(x then b(x = ( f ss β (x 1 J ss β (x β 1 D(x ln f ss β ss (x, which is valid for any (1 with a stationary fβ (x [36, 37]. In general, when β, the existence and characterizations of the limits of µ β (ω = f ss ω β (xdx and ϕ β (x = β 1 ln fβ ss (x are highly non-trivial. 3.2 Nonlinear dynamics j(x The unbalanced stationary circulation in a MB equilibrium J ss β (x = j(xeβϕ(x has a clear deterministic, underlying nonlinear dynamics dx = j(x, j(x = 0. (19 Zero divergence of the vector field j(x means the dynamics is volume preserving in phase space. Furthermore, ϕ(x is one conserved quantity: d ϕ( x(t = ϕ(x ( dx = ϕ(x j(x = 0. (20 Therefore, the dynamics in Eq. 19 is akin to a Hamiltonian system. There is an agreement between the stochastic thermodynamics and the nonlinear dynamics. Indeed, the operator decomposition in Sec. 2.3, L = L S + L A matches the vector field decomposition b(x = j(x D(x ϕ(x: 3.3 Entropy production [ ] L S u ] = [β 1 D ln (u(xe βϕ(x u(x, (21 [ ] L A u = (j(xu(x. (22 It has been shown that for a diffusion processes with MB equilibrium, the entropy production rate that is consistent with both known physics and the stochastic trajectory-based mathematical 10

11 formulation based on time reversal is the free energy decreasing rate, or non-adiabatic entropy production [8]: df (t ( = J(x, t ϕ(x + β 1 ln f(x, t dx (23 R n = µ(x, td(x µ(x, tf(x, tdx, (24 R n in which again µ(x, t = ϕ(x + β 1 ln f(x, t, as in Eq. 8. We note that even though J(x, t = j(xf(x, t f(x, td(x µ(x, t contains the conservative current j(x, it has completely disappeared in the final entropy production formula (24. Secondly, the rate of mean energy change: d ϕ(x = J(x, t ϕ(xdx (25 R n = β 2 R n ( e βµ(x,t D(x ( e βϕ(x dx. (26 The meanings of the two quantities E in (t and e p (t are yet to be elucidated for the underdamped systems. The mathematical expression for stationary E in and e p in (11 and (12 now reads R n j(xd 1 (xj(xe βϕ(x dx. (27 It has a resemblance to kinetic energy; one chould argue that in an under-damped thermal mechanical equilibrium, kinetic energy comes in and heat goes out. 3.4 Three examples We now give three examples of diffusion processes, with increassing generality, that have an MB equilibrium [8]. Ornstein-Uhlenbeck process. As a Gaussian Markov process, the Ornstein-Uhlenbeck (OU process is the most widely used stochastic-process model in science and engineering [38, 39]. Interestingly, its stationary process is always an MB equilibrium. Realizing that stationary OU process is the universal theory for linear stochastic dynamics [40, 41, 42], this result will have far-reaching implications. 11

12 An OU process has a constant diffusion matrix D and a linear b(x = Bx. All the eigenvalues of B are assumed to have positive real parts. The stationary probability density and circulation can be exactly computed in terms of the covariant matrix Ξ of a Gaussian distribution: f ss (x = ( 2π ( ( n det(ξ exp 1 2 xt Ξ 1 x, (28a J ss (x = ( B DΞ 1 xf ss (x, (28b BΞ + ΞB T = 2D. (28c J ss (x = 0 if and only if BD = DB T [43]. Then Ξ = B 1 D. In general, the solution to the Lyapunov matrix equation (28c has an integral representation Ξ = 2 0 e Bs De BT s ds. Noting that ΞB T D = D BΞ is anti-symmetric [44], ( (B J ss (x f ss (x = T DΞ 1 xf ss (x f ss (x ( 2 ( = f ss (x x ( T B DΞ 1 T Ξ 1 x ( 2x ( = f ss (x T Ξ 1 ΞB T D Ξ 1 x = 0. In fact, the linear vector field Bx, which has a decomposition ( DΞ 1 B x DΞ 1 x as in (18, can be further represented as [44] Bx = ( A + D ( 1 2 xt Ξ 1 x. (29 in which matrix A + D has a symmtric part D and an anti-symmetric part A = BΞ D. In other words, any linear vector field b(x = (A + D ϕ(x where ϕ(x is quadratic. The corresponding linear stochastic differential equation dx(t = (A + D ϕ(x + ( 2D 1 2 db(t, then, can be re-written as MdX(t = ϕ(x + ΓdB(t, (30a in which M = (A + D 1, Γ = M(2D 1 2, and ΓΓ T = 2MDM T = M + M T, (30b 12

13 which is twice the symmetric part of M. Stochastic differential equations expressed such as (30, which describes a stochastic dynamics without detailed balance but still reaching an MB equilibrium e ϕ(x, first appeared in [45]. Klein-Kramers equation. The Klein-Kramers equation is the canonical stochastic Newtonian dynamics with a stochastic damping that satisfies fluctuation-dissipation relation, thus a Maxwell- Boltzmann distribution in its stationary state. It has D = k B T ( η(x, b(x, y = ( m 1 y U (x m 1 η(xy, (31 and f ss (x, y = exp ( H(x,y k B T where the Hamiltonian function H(x, y is the total mechanical energy and H(x, y = y2 + U(x, (32 2m J ss (x, y = ( m 1 y U (x f ss (x, y, j(x, y = ( H/ y H/ x. (33 d (x, yt = j(x, y is indeed the underlying Hamiltonian dynamics. The stationary stochastic circulation J ss (x is the Hamiltonian conservative dynamics weighted by the stationary probability f ss (x, y. P. Ao s stochastic process. P. Ao and his coworkers have generalized the Eq. 29 and the Kramers-Klein equation to a class of nonlinear diffusion processes [44, 45, 46, 47] with: D(x = 1 2 ( G(x + G T (x, b(x = G(x ϕ(x. (34 The stationary probability density and circulation are again readily obtained: f ss (x = e ϕ(x, J ss (x = 1 2 ( G(x G T (x f ss (x. (35 Obviously, J ss (x f ss (x = 0. Three interesting mathematical questions arise from this model: (i For any divergence-free vector field j(x with a first integral ϕ(x, e.g., j(x ϕ(x, whether there always exists an anti-symmetrix A(x such that j(x = A(x ϕ(x? This is true for any Hamiltonian system, with even n, where A is symplectic. For n = 3, such a j(x is called a gradient conjugate system, whose solution curves are all closed orbits [48]. 13

14 (ii For a vector field b(x that satisfies the decomposition in (18, what is the relationship between a Sinai-Bowen-Ruelle type invariant measure, when it exists, and the the e βϕ(x in the limit of β 1 0 [49]? (iii For any vector field b(x, whether there always exists an symmetric matrix D(x such that b(x = j(x D(x ϕ(x with diverence-free j(x and j(x ϕ(x? 4 Helmholtz theorem and Carnot cycle We now consider the family of diffusion processes in terms of the parameter α in (1. All the discussions in Sec. 2 and Sec. 3 have been focused on the dynamics and thermodynamics of one autonomous dynamical system, or time-homogeneous stochastic process, with a single α value. Particularly, we have noticed that the generalized, nonequilibrium free energy F (t in (10 takes its stationary state as the reference point. Classical thermodynamics, however, is a theory of relationships among different stationary states connected through changing a parameter. Being able to provide a common energy reference point for diffusion processes with different α, one needs a unique ϕ(x, α which can not be obtained unambiguously from the analysis presented so far in Sec Generalizing Helmholtz theorem One should recognize that the long-time state of x(t following the equation of motion (19 is not a single point in R n ; but a bounded orbit confined in a particular level set of ϕ; it is actually a state of stationary motion [50]. This distinguishes a microscopic state in classical mechanics from a macroscopic state in classical thermodynamics. In general, x(t is not ergodic on the entire level set. However, realizing that j(x is only a deterministic representation of the stochastic circulation, one expects a time-scale separation between the intra-ϕ-level-set motion and motion across level sets [51]. To represent and characterize an entire ϕ level set, Boltzmann s idea was to quantify it using some geometric quantities. The volume it contains in R n is one example: ( σ B (h, α = ln. (36 dx ϕ(x,α h 14

15 An elementary calculation of probability yields β 1 ln e βϕ(x,α dx = ϕ(x, α β 1 ln h<ϕ(x,α h+dh h<ϕ(x,α h+dh dx. (37 Since free energy is equal to internal energy minus temperature times entropy, recognizing the last term in (37 as entropy is almost mandatory. Then, ( Z α (β = e βϕ(x,α dx = e βh+σ σb B(h,α dh (38 R n h α ( = e βh ds dh. (39 x ϕ(x, α ϕ(x,α=h The term inside the parenthesis in (39 is known as microcanonical partition function [52]. Eq. 39 relates the microcanonical partition function to the canonical partition function Z α (β. We now apply the Helmholtz theorem [50] for the family of conservative dynamics j(x, α with parameter α, which has a conserved quantity ϕ(x, α [51]. Note that due to its definition, the vector field j(x, α is structurally stable within the framework of the general diffusion process (1. σ B in (36 is an increasing function of h, thus implicit function theorem applies to σ B (h, α. Thus we have h = h(σ B, α, and ( h dh = σ B in which ( σb h α = = ϕ(x,α=h ϕ(x,α=h dσ B + α ds x ϕ(x, α ϕ(x,α h dx ( h dα, (40 α σ B ( ϕ(x, α ds x k x k x j,j k,α x ϕ(x, α ds x ϕ(x, α ϕ(x,α=h 1 (41, (42 which is the same for all k = 1, 2,, n. This is known as the virial theorem in classical mechanics. Gauss-Ostrogradsky-Green s divergence theorem is used to derive (42 from (41. We shall denote θ(h, α ( 1 ( σb h, F α (h, α. (43 h α α σ B 15

16 Then, F α θ = ( h α σ B ( σb = h α ( σb. (44 α h Historically, the significance of the Helmholtz theorem is generalizing mechanical energy conservation to the First Law of Thermodynamics. It provided a mechanical theory of heat [50]. In the present theory, our Eq. 40 has extended conservative ϕ(x, α defined for each stochastic dynamical system with a particular α to a much broader energy conservation law h(σ B, α among the entire family of dynamical systems with different α. Free energy and entropy of a Gaussian distribution. For a n-dimensional Gaussian distribution with covariance matrix Ξ(α, the quadratic potential function ϕ(x has the form 1 2 n k=1 λ 1 k ξ2 k under an orthonormal transformation, where λs are the eigenvalues of Ξ. Then ( n σ B (h = ln 2hλk = n 2 ln h ln det ( Ξ + n 2 ln 2 + ln V n, (45 V n k=1 in which V n = π n/2 Γ 1( n is the volume of an n-dimensional Euclidean ball with radius 1. Free energy of the same quadratic ϕ(x, according to canonical partition function, is ( β 1 ln exp β { n R 2 xt Ξ 1 x dx = β 1 2 ln ( 2πβ ln det ( Ξ }. n Mean internal energy is h = n, and entropy is 2β n 2 ln h ln det ( Ξ + n 2 + n ( 4π 2 ln, n which agrees with the σ B (h, α in (45 when n is large, via Stirling s formula. If the determinant of Ξ(α is linearly dependent upon a parameter α, then, ( σ B α h = (2α 1, and αf α = 1 2 θ. One could identify θ with temperature, α as volume, and F α as pressure, then this relation is the law of ideal gas in classical thermodynamics. Eqs. 43 and 45 also give θ = 2h/n. h being a function of θ alone is known as Joule s law, which states that the internal energy of an ideal gas is a function only of its temperature. Based on these observations, it is not unreasonable to suggest the OU process as a mesoscopic dynamic model for stochastic thermodynamic behavior of an ideal system, such as ideal gases and ideal solutions [53]. 16

17 β as an ensemble average of ( σ B / h α. be re-written into a different expression: Note that the right-hand-side of (38 can also Z α (β = in which β plays the role of ( σ B h α average of ( σ B : h α β = 1 Z α (β e βh+σ B(h,α βdh, (46 in (38. β, in fact, can be expressed implicitly as an ensemble e βh+σ B(h,α ( σb dh. (47 h α 4.2 Carnot cycle in a family of diffusion processes with MB equilibrium Functional relations among triple quantities such as ( σ B, h, α, ( σ B, F α, α, and ( θ, F α, α are called equations of state in the classical thermodynamics. They are powerful mathematical tools quantifying long-time behavior of a family of conservative dynamics j(x, α. A Carnot cycle consists of two iso-θ curves and two iso-σ B curves in α versus F α plane. Let us again consider the simple model σ B (h, α = µ ln h + ν ln α. Then we have equations for iso-θ curves and iso-σ B curves: αf α = νθ, and α 1+ν/µ F α = ν µ eσ B/µ. (48 5 Externally driven cycle of a diffusion process with dissipative circulation The foregoing discussion clearly points to two mathematical objects J ss (x and ϕ(x = β 1 ln f ss (x associated with a stationary diffusion process. We shall set β = 1 in the following discussion. The steamlines of vector field J ss (x and the level sets of ϕ(x are perfect matched in a Maxwell- Boltzmann equilibrium with non-dissipative circulation. If the condition (18 is not met, then a diffusion process has stationary circulation with dissipation. This implies the process also has to be externally driven. Thus, its stationary process is in a nonequilibrium steady state, as illustrated in Fig. 1(A. See [54] for a recent paper on the subject. One can in fact idealize any closed orbit in phase space into four pieces as shown in Fig. 1(B: Movement from c d is a spontaneous relaxation from high ϕ-level to low ϕ-level accompanied 17

18 b c a d (A (B Figure 1: (A Diffusion processes that have non-orthogonal J ss (x and ϕ(x have the streamlines of J ss passing through different level sets of ϕ. Being a conservative dynamics, the streamlines of vector field J ss have almost closed orbits according to Poincaré recurrence theorem. Any closed orbit can be approximated by portions that are confined in ϕ-level sets, and portions that perpendicular to ϕ-level sets. (B An idealized red closed loop abcda consists four steps. Assuming ϕ 1 > ϕ 2, then step ab decreases in probability, thus it in general has to be externally driven; steps bc and da are confined in ϕ-level sets, thus they are conservative; step cd increases in probability, thus it is spontaneous with dissipation. with dissipation; it is followed by a non-dissipative step from d a confined in the level set ϕ(x = ϕ 2 ; then followed by a transition a b from low ϕ-level to high ϕ-level, which has to driven by an external force; and finally another non-dissipative move from b c confined in the level set ϕ(x = ϕ 1. The analysis of energetic steps discussed above, and illustrated in Fig. 1, is analogous to a pendulum system with damping and being driven: m d2 x 2 = k sin x η ( dx + ξ(t, (49 in which ξ(t is an oscillatory driving force. In the absence of damping ηẋ and driving force ξ(t, the mechanical energy level sets are H ( x, ẋ = m ( 2 ẋ2 + k 1 cos x. Then, d H( x, ẋ = ηẋ 2 + ẋξ(t. (50 18

19 When the right-hand-side of (50 is positive, the system gains energy; and when it is negative, the system dissipates energy. Over a complete cycle, these two terms have to balance with each other. We also noticed that they are even and odd functions of ẋ, respectively. One could argue that the orthogonal relation between j(x and ϕ(x, for all x, in a system with MB equilibrium is a local equilibrium condition. Such a condition provides a mathematical basis for organizing the entire R n state space, and the conservative motions, in terms of a single scalar ϕ(x. It has been suggested as a possible mathematical statement of the Zeroth Law of thermodynamics [8]. In contrast, the system in Fig. 1 has its level sets and streamlines out of equilibrium. 6 Discussion One interesting implication of the present work, perhaps, is that a linear stochastic dynamics is always consistent with a Maxwell-Boltzmann equilibrium, together with a Hamiltonian system-like conservative dynamics [44, 8]. This result completely unifies the stochatsic thermodynamic theory and the linear phenomenological approaches to equilibrium fluctuations pioneered by Einstein and Onsager, and extended by many others, e.g., R. Kubo, Landau-Lifshitz, H. B. Callen, M. Lax, and J. Keizer, to name a few. It is important to realize, therefore, that near a stable dynamic fixed point, one can not determine the nature of equilibrium vs. nonequilibrium fluctuations in a subsystem from the internal data alone. Additional information concerning the external environment is required to uniquely select one of the two possible thermodynamics. In fact, the origin of the entropy production in an overdamped nonequilibrium steady state is outside the subsystem, as the notions of source and sink clearly imply. Therefore, a nonequilibrium steady state of a subsystem can only be fully understood by including its envirnment; and for a universe without an outside, a underdamped thermodynamics with the heat death is the only logic consequence. We have recently suggested that the mathematical description of stochastic dynamics is an appropriate analytical framework for P. W. Anderson s hierarchical structure of science [21]. The origin of randomness has been widely discussed by many scholars; for example Poincaré has stated in 1908 that [55] A very small cause, which escapes us, determines a considerable effect which we 19

20 cannot ignore, and we then say that this effect is due to chance. In the light of [1, 2, 3], this might be updated: A very complex collection of causes, which are not understood by us, determines an un-avoidable consequence which we cannot ignore, and we then say that this effect is due to chance, or our ignorance. The mathematical approach is complementary to other phenomenological investigations that have gone beyond classical equilibrium thermodynamics. Two particularly worth mentioning theories are steady state thermodynamics [56] and the finite time thermodynamics [57]. The former has been shown to be consistent with stochastic processes with either discrete or continuous state spaces [30, 58]; and the latter considered the notion of thermodynamic dissipation, especially in the coupling between a system and its baths, without actually requiring a time-dependent description. The stochastic diffusion theory presented in the present work could provide a richer narrative for complex phenomenon which has a stochastic dynamic description, but currently lacks a concrete connection to vocabularies with mechanical and statistical thermodynamic implications, for example behavioral economics. Paraphrasing Montroll and Green [59]: The aim of a stochastic thermodynamic theory is to develop a formalism from which one can deduce the collective behavior of complex systems composed of a large number of individuals from a specification of the component species, the laws of force which govern interactions, and the nature of their surroundings. Since the work of Kirkwood [9], it has become clear that the laws of force can themselves emergent perperties with statistical (entropic nature. Indeed, Eq. 40 could be recognized as one of such. Probability is a force of nature. In the western legal system, this term refers to an event outside of human control for which no one can be held responsible. Still, something no individual, or a small group of individuals, can be held responsible is nevertheless responsible by each and every individual, together. More is different. References [1] Anderson P W 1972 More is different: Broken symmetry and the nature of the hierarchical structure of science Science

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